Calculation of limit with different root indexes

Question

maybe you can help me to find a limit of a function with different root indexes? This type of roots is pretty hard for me and I want to know the pattern how to solve exercises like this. Thanks!

$$\lim_{x \to \infty} \frac{x^{1/3} - 2x^{3/2}} {x^{4/5} + x^{3/2}} $$

Answer 1

If you don't feel comfortable with roots, get rid of them.

For $x>0$, $$ \frac{x^{1/3} - 2x^{3/2}} {x^{4/5} + x^{3/2}} = \frac{x^{10/30} - 2x^{45/30}} {x^{24/30} + x^{45/30}} $$ Now, setting $y=x^{1/30}$, we have $\lim_{x\to\infty} y = \infty$, so that when we substitute we get $$ \lim_{x\to\infty}\frac{x^{10/30} - 2x^{45/30}} {x^{24/30} + x^{45/30}} = \lim_{y\to\infty}\frac{y^{10} - 2y^{45}} {y^{24} + y^{45}} $$ Do you know how to figure out this last one?

Detailed solution:

factor both numerator and denominator by the largest term, i.e., $y^{45}$.$$\frac{y^{10} - 2y^{45}} {y^{24} + y^{45}} =\frac{ - 2 + y^{-35}} {1+y^{-21}} \xrightarrow[y\to\infty]{}\frac{-2+0}{1+0}$$